ACTED ON BY NO EXTERNAL FORCES, 779 



called is not to be confounded with the intersection above named, but is the mutual 

 intersection of two ideal instantaneous orbits with each other and the invariable plane. 



In every ordinary dynamical problem, by a well-kno\vn simple contrivance, the time 

 element may be preliminarily thrown out of the differential equations of the motion ; in 

 the class of which the three noble and celebrated questions here referred to are the 

 conspicuous types, a certain space element is capable of being similarly left out to the 

 end ; thus the number of linear differential equations required for the determination of 

 the remaining elements is reduced by two, and if all the integrals of this reduced system 

 are capable of being found, then we know, a priori, by the theory of the last multiplier, 

 how to reduce to quadi-atures the values of the two outstanding elements. The process 

 whereby the space coordinate referring to absolute position is, so to say, avoided in this 

 class of dynamical questions, is not, or at least need not be considered as, one of elimi- 

 nation properly so called ; elimination is the act of extruding a variable from a system 

 of equations in which it has appeared ; the process to be applied in the case before us 

 is one not of extrusion, but of exclusion ah initio, or as it may be rendered in a single 

 word, of ab-limination. 



I propose at an early moment to return to a consideration of the particular method 

 of ab-limination above indicated as applicable to the problem of three bodies, in the 

 study of which this memoir took its rise. 



